Abstract
An improved Fourier series method (IFSM) is applied to study the free and forced vibration characteristics of the moderately thick laminated composite rectangular plates on the elastic Winkler or Pasternak foundations which have elastic uniform supports and multipoints supports. The formulation is based on the first-order shear deformation theory (FSDT) and combined with artificial virtual spring technology and the plate-foundation interaction by establishing the two-parameter foundation model. Under the framework of this paper, the displacement and rotation functions are expressed as a double Fourier cosine series and two supplementary functions which have no relations to boundary conditions. The Rayleigh-Ritz technique is applied to solve all the series expansion coefficients. The accuracy of the results obtained by the present method is validated by being compared with the results of literatures and Finite Element Method (FEM). In this paper, some results are obtained by analyzing the varying parameters, such as different boundary conditions, the number of layers and points, the spring stiffness parameters, and foundation parameters, which can provide a benchmark for the future research.
1. Introduction
A variety of composite plates on the elastic foundations are widely used in civil engineering, like rigid road plate, airport pavement, building foundations, dock platform, and so on. It is of great significance for design, use, and maintenance to study the mechanical properties of composite plates on elastic foundation. Therefore, the study on the vibration characteristics of laminated plates on the elastic foundations has been one of the hot spots in recent years. According to the existing research, the elastic foundational plate models can be divided into two categories which are Winkler-type foundation and two-parameter foundation.
Winkler foundation is the first proposed linear elastic theoretical model. Laura et al. [1] solved the free vibrations of a circular thin plate on Winkler foundation with varying thickness by using linear analysis and the Rayleigh-Schmidt method. Liew et al. [2] extended an approximate analysis of rectangular Mindlin plates resting on Winkler foundations based on the differential quadrature method (DQM). The plates were subject to arbitrary combination of free, simply supported, and clamped boundary conditions. Gupta et al. [3] discussed the effect of Winkler foundation on axisymmetric vibrations of polar orthotropic circular plates with variable thickness based on the classical plate theory. Xiang [4] investigated the effect of multisegment Winkler foundations on the rectangular Mindlin plates. They divided the four sides into two groups, in which the two parallel edges were one group. A set of edges were simply supported and another group was the combinations of uniform and classical boundary conditions. Younesian et al. studied strongly nonlinear generalized duffing oscillators [5] by using He’s frequency–amplitude formulation and He’s energy balance method and then proposed a closed form expression for the dynamic response of an elastic plate rested on a nonlinear elastic Winkler foundation [6]. Ansari et al. investigated the forced vibration of microbeam structures supported by nonlinear viscoelastic-type foundation [7], Kelvin–Voight foundation [8], and Winkler and Pasternak foundations [9] based on the Galerkin approach and multiple time-scales method. There are many related researches on the Winkler foundation, and we can refer to the relevant refs. [10–12]. The calculated results of the Winkler-type foundation plate model are very different from the actual results. In some special conditions, especially, the results could not meet the requirements any more by using the Winkler foundation model.
With the deepening of the research, the two-parameter elastic foundation model is proposed. Two independent parameters are used to represent the compressive strength and shear strength of the soil, which can overcome the inherent defects of the Winkler foundation model and effectively eliminate the discontinuity. In the two-parameter model, the Pasternak foundation is widely used. Nedri et al. [13] studied the free vibration analysis of simply supported thick laminated rectangular composite plates resting on elastic Pasternak-type foundations based on refined hyperbolic shear deformation theory. Baltacıoğlu et al. [14] used the discrete singular convolution method to realize nonlinear static analysis of a thick laminated rectangular composite plate on nonlinear foundation by using the FSDT. Singh et al. [15] dealt with the postbuckling shear deformation of the laminated composite plates on the Pasternak-type foundation on the basis of HSDT and random system properties. It is the first time that a perturbation technique has been successfully combined with direct iterative technique by neglecting the changes in nonlinear stiffness matrix due to random variation of transverse displacements during iteration. Malekzadeh et al. [16] investigated the effect of nonideal simple supports and initial stresses on the vibration of laminated rectangular plates on Pasternak foundation based on the Lindstedt-Poincare perturbation technique. Kutlu et al. [17] derived a mixed-type finite element formulation to study the dynamic response of the Mindlin plate–arbitrarily orthotropic Pasternak foundation interaction by applying the Gâteaux differential. Tornabene et al. [18] considered the static and dynamic analyses of laminated doubly curved and degenerate shells and panels on the Winkler and Pasternak foundations by using the generalized differential quadrature (GDQ) method and FSDT. Jahromi et al. [19] studied free vibration of moderately thick rectangular plate partially resting on Pasternak foundation based on GDQ and FSDT. The boundary conditions of plate were considered as combinations of free, simple, or clamped support. Li and Zhang [20] investigated free vibration analysis of magnetoelectroelastic plate resting on a Pasternak-type foundation based on Mindlin plate theory. Khalili et al. [21] used the Lindstedt-Poincare perturbation technique to study the buckling of nonideal rectangular laminated plate on Pasternak foundation. One edge of the plate was allowed a small nonzero deflection and movement. Dehghany and Farajpour [22] dealt with exact solution for free vibration analysis of simply supported rectangular plates on Pasternak foundation on the basis of three-dimensional elasticity theory. Briscoe et al. [23] presented a solution for the buckling strength of simply supported plates on the Pasternak-type foundation under in-plane bending loads by using the minimum potential energy principle. Thai et al. [24] proposed a simple refined theory for studying bending, buckling, and vibration of thick plates resting on Pasternak foundation. The boundary conditions of rectangular plates were expressed as two opposite edges with simply supported plates and the other two edges with arbitrary boundary conditions. Idowu et al. [25] proposed a fourth-order partial differential equation to study the dynamic effects of viscous damping on isotropic rectangular plates resting on Pasternak foundation subjected to moving loads. Bahmyari and Khedmati [26] used shear deformable plate theory and Element-Free Galerkin Method to study the free vibration analysis of nonhomogeneous moderately thick plates with point supports resting on Pasternak elastic foundation. Kiani et al. [27] considered instability of simply supported sandwich plates with functionally graded material (FGM) face sheets resting on the Pasternak foundations based on FSDT. There have been a lot of literatures of the plate resting on the Pasternak-type foundation. We can also check the relevant references, such as [28, 29]. However, these literatures have great limitations for the moderately thick or thick laminated rectangular plates resting on the elastic foundation which have arbitrary and special boundary conditions. It is necessary to study the effects of boundary conditions on the free and forced vibration characteristics of plate resting on the elastic foundation.
Stimulated by the restriction of the plate boundary conditions in the existing researches, the free and forced vibration characteristics of the moderately thick laminated rectangular plates are analyzed which rest on the Winkler or Pasternak foundations and have various uniform supported and multipoints supported boundary conditions. It should be pointed out that the free vibration analysis of the moderately thick laminated composite rectangular plate with nonuniform boundary conditions [30] has been done previously. However, it only discussed the free vibration analysis of the laminated plate with partial supports and multipoints supports. In this proposition, the free and forced vibration analyses of uniform and multipoints supported laminated rectangular plate resting on the Winkler and Pasternak foundations are studied in this paper. As we all know, there are many structural forms of plate in engineering applications, like stiffened plates [31–34], cracked rectangular plates [35–38], corroded plates [39–43], and so on. As the most common basic model, the rectangular plate structure is widely studied. An IFSM is extended to study the free and forced vibration of rectangular plates on the elastic Winkler-type and Pasternak-type foundation. This method is previously studied by Zhang and Li [44, 45]. According to the FSDT, the five displacement functions can be written as feasible period superposition functions. Their specific expressions are a double Fourier cosine series and two supplementary functions by ignoring the influence of boundary conditions. On the basis of the traditional Fourier series, these supplementary functions are added to eliminate the discontinuous or jumping phenomenon in the boundaries which are regarded as a periodic function and defined within the entire coordinates of laminate plate. These unknown coefficients are defined in the generalized coordinates which can be solved by Rayleigh-Ritz procedure. It is very easy to realize the change of different boundary conditions by changing the stiffness value of the five springs on the four edges. The results obtained by this method are compared with those results obtained by literatures and FEM, which show good agreement. The work mainly deals with the consequences of the practical and significant constraints such as various uniform supports, multipoints supports, and various values of the foundation parameters.
2. Theoretical Analysis
A combination technique is used to get the vibration characteristics of orthotropic laminated plate structure based on artificial spring technique and Rayleigh-Ritz method. This plate under uniform supports or multipoints supports is rested on elastic Winkler-type and Pasternak-type foundation.
2.1. Establishment of the Model
As shown in Figure 1, a laminated rectangular plate model is established to analyze vibration characteristics. For the plate, the length, width, and thickness are , , and . Establish a coordinate system in the plate mid-surface. In this coordinate system, the , , and represent the length, width, and thickness directions of the studied plate. Symbol is the laying angle between the layer fiber direction and the -axis. For the sake of brevity, it is supposed that every lamina has the same material properties and thickness. For the elastic foundation, and are defined as linear Winkler foundation and linear Pasternak foundation parameters. Five types of spring are used to describe the boundary conditions of laminated foundation plate, which are linear springs , , and and rotational springs (, ), respectively. The arbitrary boundary conditions can be realized by setting the stiffness values of the five different springs [46–50]. For instance, the free boundary condition can be easily gained when the spring stiffness values on the four edges are zero. All the stiffness values are set to a large value to achieve the clamped boundary conditions in the numerical calculation.
2.2. Relationship between Kinematics and Stress
According to the displacements and rotations of the middle surface for the established plate model, the displacements can be expressed based on FSDT [51–53].where , , and represent the middle surface displacements of the plate in the , , and directions and and are the rotations of transverse normal for - and -axes, respectively. Then is the time variable.
For rectangular laminated plates, the linear strains-displacement relations of ’th layer can be got according to the strain–stress relationship of elasticity theory
According to the above formulas, , , and express the normal and shear strains in the coordinate system. The symbols of transverse shear strains are marked as and which can be regarded as constant and ignore the thickness change. , , and represent the corresponding curvature and twist changes. Besides, expresses the thickness variable range of ’th layer. Expressions for strain and generalized displacement relations can be established as
According to the generalized Hooke’s law, the relation between stress and strain can be expressed asin which and are the normal stresses and , , and are shear stresses in the coordinate system. Besides, the specific expression of ’th layered stiffness coefficients [54, 55] can be written aswhere is the laying angle and represents the material coefficients of the ’th layer which can be obtained by building the relationships with longitudinal modulus , the transverse modulus , Poisson’s ratios and , and shear moduli , , and .
In order to seek simplicity, the material constants of each layer are the same, which can be expressed as , , and . In addition, the relationship of and is . It should be pointed out that the isotropic plate resting on the elastic foundations can be easily analyzed by letting , . By introducing the shear correction factor , the relation between the generalized forces and strains can be finally obtained [51, 52]. The specific expressions are shown as follows:in which the generalized forces include normal forces (, ), shear force (), bending moments (, ), twisting moment (), and transverse shear forces (, ). In addition, , , and are extensional, extensional-bending, and bending stiffness coefficients, whose expressions are written asin which donates the total number of layers. Moreover, when we study the vibration characteristics of the isotropic plate, the extensional-bending stiffness coefficients are zero and and are independent.
2.3. Energy Equation
The main work of this paper is to investigate the vibration characteristics of the moderately thick laminated composite rectangular plate with uniform and multipoints supports which is rested on the elastic foundations. Rayleigh-Ritz energy method is extended to study the free and forced vibration of plate on the Winkler and Pasternak foundations. The Lagrangian energy function for the laminated plate can be written as
expresses the total kinetic energy of laminated plate whose expression isin which , , and are the inertia terms of the plate, and is area density of the ’th layer.
expresses the strain energy for the moderately thick plates:
Substituting (3), (7), and (8) into (11) can obtain the relations between strain energy and displacements in mid-surface of plate. Then, strain energy expression can be written as a superposition of three components, which are stretching energy (), bending energy (), and bending–stretching coupling energy ().
The strain energy due to the Winkler and Pasternak foundations is given by
For the elastic Winkler foundation, we just need to set the Pasternak foundation parameter KS to zero. Then, the influence of the free vibration for the plate which is on the Winkler-type elastic foundation can be studied with the change of the Winkler foundation parameter KW. In this paper, the main interests are studying the special boundary constraints and variation of Winkler and Pasternak foundations by introducing artificial virtual spring technology, which can be found in [56–58]. So the cases of uniform supported and multipoints supported boundary conditions whose plates are rested on the Winkler and Pasternak foundations will be considered here.
is the potential energy on the four edges of the plate with elastic uniform boundary conditions. It is simulated with five kinds of springs evenly distributed on four edges.
For the cases of multipoints supports, the supported points are evenly distributed on the four edges. For intuitive understanding, Figure 2 gives the distributions of 4-point, 8-point, and 16-point supported boundary conditions. They can be regarded as the discretization of uniform support. Therefore, the potential energy for the plate with multipoints supports can be written as
(a) 4-point support
(b) 8-point support
(c) 16-point support
expresses the work done by the external excitation force on the moderately thick plates:where is the external load distribution function on the plate. In this paper, the normal harmonic point force on the plate is applied to study the flexural vibration behavior of the composite plate. Therefore, the load distribution function can be expressed aswhere (, ) is the position of point force. is the 2D Dirac function.
2.4. Kinematics Balance Equation
The kinematics balance equations for the moderately thick plate can be obtained by Hamilton’s principle [51, 52].
As we can see from (18), the displacement functions required second-order derivatives. The kinematics balance equations can be rewritten as a matrix form by combining (3), (7), and (18) simultaneously.where linear differential operators are expressed as and . The specific expressions are given below:
2.5. Displacement Expression
An IFSM is applied to express the displacements of the moderately thick laminated rectangular plate on the elastic foundations. The expressions ignore the boundary conditions and eliminate the discontinuous or jumping phenomenon in the boundaries. They can be regarded as a periodic function which is defined within the entire coordinates [59]. The specific displacement expressions of the plate are given below:where and . Besides, , , , , and are 2D Fourier coefficients vector, which are composed by , , , , and , respectively.
There have been a lot of references about the Rayleigh-Ritz technology, such as [60–62], and Ritz like methods, such as [63–65]. In this paper, based on the Rayleigh-Ritz technology, a set of linear algebraic equations directed against Fourier unknown coefficients on displacement equations can be got by combining (18)–(23e) simultaneously. A matrix form can be obtained by transformation
When we study the free vibration of the plate, we only need to ignore the external force vector . , , and are stiffness matrix, mass matrix, and unknown Fourier coefficients vector, separately. The specified expressions of , , and are listed in the Appendix. It should be pointed out that is a combination of the five 2D Fourier coefficients vectors, whose form is
Form (24), we can easily find that the circular frequency is the square root of eigenvalues and is the eigenvectors. When we study the vibration response of the elastic supported plates resting on the elastic Winkler or Pasternak foundations, can be obtained by giving a frequency value. Then, the exact displacements expressions of plate can be got by putting into (21)–(23e). In order to explain the innovation of this paper more intuitively, a brief illustration is given in Figure 3. Compared with [30, 66], this paper has three main original ideas. Firstly, this paper focuses on the effect of elastic foundation on the plate. Secondly, the displacement expressions of the plate are improved. Thirdly, forced vibration response of plate on the elastic foundation is studied by introducing a harmonic point force. In addition, the corresponding results are also given in the next section.
3. Results and Discussions
In this section, some calculation cases involving symmetrically and antisymmetrically laminated plates resting on the elastic Winkler and Pasternak foundations are considered, which have various uniform or multipoints supported boundary conditions. Comparisons are made with the available results of literatures and FEM. For the sake of brevity, the letters F, S, and C are used to represent completely free, simply supported, and clamped edges. In addition, a simplified combination of letters (in Table 3) is used to characterize the boundary condition of the antisymmetrically laminated plates. For example, the SCSC shows that the laminated plates have S, C, S, and C at the four edges of , , , and , respectively. Besides, material properties of the elastic lamina are supposed as Material I: ; ; ; ; Material II: ; ; ; ;
3.1. Convergence Analysis
For the displacement admissible functions constructed in (21)–(23e), the appropriate truncation value should be set in numerical evaluation. The size of the values of and is the direct representation of the convergence of the method. Table 1 shows the convergence of the first eight frequency parameters of the angle-ply (0/90°) rectangular laminated plates on the elastic Winkler foundation. The geometrical dimensions are , , the material type is Material I, and the foundation parameters are and .
From Table 1, we can find that the present method shows good convergence. For example, for the free plate on the Winkler foundation, the biggest difference for the worst case which is made is the contrast of 8 × 8 and 18 × 18 being less than 0.13%, while the biggest difference for the worst case which is made is the contrast of 14 × 14 and 22 × 22 being less than 0.018%. It is not difficult to find that the biggest difference for the worst case which is made is the contrast of 18 × 18 and 22 × 22 being zero. In order to make the results more accurate, all the truncated values of and are 18 in the next calculations.
3.2. The Uniform Supported Plate on Elastic Foundations
In this section, free and forced vibration characteristics of the thin and moderately thick plate are discussed. The accuracy of the present method is verified by being compared with the results of Shen et al. [67]. The first frequency parameters of a three-layered square plate which are simply supported and rest on different elastic foundations are displayed in Table 2. From the table, we can see that the first frequency increases when the two kinds of foundation parameters and the length-thickness ratios increase. In addition, this method is also applicable to the analysis of thin plates. In Table 3, the first six frequency parameters of the two angle-ply (0/90°) clamped square thin plates on the Winkler foundation are given. The geometry parameters of the plate layers are , , and the material type is Material II. The results obtained by the presented method are in good agreement with the FEM results.
For laminated plate on elastic Pasternak foundation, it is significant to study the change of natural frequency under various boundary conditions. Table 4 shows the first three frequency parameters for the two-layered [+45°/−45°] rectangular plate. It lists the change of of the plate which rests on the elastic Pasternak-type foundation. There are six kinds of boundary constraints, that is, FFFF, FSFS, FCFC, SSSS, and CCCC. Five kinds of anisotropic ratios are given, which are 10, 20, 30, 40, and 50. In addition, two foundation parameters are and , respectively. An interesting phenomenon is that the increase of anisotropic ratio has little effect on the natural frequency. But the frequency parameters increase in general with the binding force of the BC increasing as exhibited in Table 4.
The effects of the boundary spring stiffness on free vibration behaviors of the laminated plate are investigated based on parametric study, which can deepen the understanding. Five elastic boundary restraint ratios are = , , , , and , that is, ; ; ; ; . is the structural stiffness coefficient, whose expression is . Figure 4 exhibits the curves of lowest three frequency parameters with the change of restraint parameters for the laminated composite rectangular plates resting on the elastic foundation. The unsymmetrical lamination scheme [45°/−45°/45°/−45°] of the composite plate is brought. The laminated plates are supported by a set of spring component on the four edges with stiffness values varying from to . The geometry constants of the plate layers are , , and the material type is Material II. Two foundation parameters are and , respectively. From the figures, it can be seen clearly that when the boundary spring stiffness parameter is less than , the change of the spring stiffness values has little effect on the natural frequency of the laminated plate. The laminated plate frequency will increase quickly with the increase of the stiffness when the value changes between and . But when the value is more than , the natural frequency of the plate is almost constant. In addition, we can also find that the greatest impact on the frequency is the spring parameters and . For and , they almost have no effect on the frequency. The interesting thing is that has no effect on the first two frequencies, while the effect on the third frequency is greater than the elastic spring parameters of and .
As mentioned earlier, the isotropic plates can be easily obtained by letting and . Figure 5 gives the transverse vibration displacement level on direction of isotropic plates in the frequency range of 0–800 Hz by exerting a harmonic point force on the simple supported plate. The amplitude of the point force is 1 N. It will be applied to the following cases of forced vibrations. The position of point force is at (, ). The positions of the two observation points are at (, ) and (, ). Material parameters of plate are N/m2, , and kg/m2. The geometry parameters of the plate are , , and . The results obtained by the present method are compared with those obtained by FEM which show good agreement. It shows that this method is also applicable to the thin plate structures. As we can see from Figure 5, the thickness of the plate has a great effect on the displacement response. The vibration displacement level increases with the decrease of the plate thickness.
(a) Displacement response at (, )
(b) Displacement response at (, )
In order to further study the effect of elastic support on the transverse vibration response of the four-layered [0/90°/0/90°] rectangular plate, three elastic boundary conditions are proposed according to Figure 4. They are , , and . The geometry constants of the plate layers are , , and the material type is Material II. Two foundation parameters are and , respectively. The position of point force is at (, ). The positions of the two observation points are at (, ) and (, ). It can be seen from Figure 6 that the values of natural frequencies increase in the frequency range of 0–1000 Hz with the increase of the boundary springs stiffness. Moreover, the vibration displacements decrease with the increase of the boundary springs stiffness.
(a) Displacement response at (, )
(b) Displacement response at (, )
3.3. The Multipoints Supported Plate on Elastic Foundations
In this section, the free and forced vibrations of laminated plates on elastic foundation with various multipoints supported boundary constraints are investigated. For this plate model, the geometrical dimensions are m, m, and the material type is Material II. In order to simplify, various constraint values of multipoints supports are set. For example, K7 expresses that the stiffness value of five kinds of spring is 107. As shown in Table 5, the first six of for the plate model under consideration with 8-point support, resting on the Winkler foundations, by using the method proposed in this paper are compared with the results of numerical calculation which use the ABAQUS model. The results of the two solutions are in good agreement with K7 elastic supported boundary conditions. Besides, there is a cognitive phenomenon that the frequency parameters increase gradually, especially after the fourth frequency, when the Winkler parameter increases.
Based on the conclusions drawn from Table 5, we have proved the accuracy of the present method. So we will directly give some frequency values of the antisymmetric laminated plate with the 8-point K7 elastic supports under the elastic Pasternak foundation. Table 6 shows some of the frequency parameters of the two kinds of cross-ply square plates, that is, angle-ply [0/90°] and [0/90°/0/90°]. This plate model is the same as the model considered in Table 4. It is not hard to find that the natural frequency increases when foundation parameters increase, and the effect of the Winkler parameter on the plate frequency is much larger than the Pasternak foundation parameter . In addition, the frequency of the four-layered plate is a little bigger than the frequency of the two-layered plate. In the following research, we will give the frequency of the plate on the elastic foundation changing with the increase of the number of layers.
In this part, we discuss the effect of boundary spring stiffness on the plate vibration. Here, we set up five kinds of elastic boundary restraint parameters , , , , and of multipoints supports, just like the parameters in Figure 4. Figure 7 exhibits the curves of lowest three frequency parameters with the change of restraint parameters of the laminated composite rectangular plates resting on the elastic foundation. The symmetrical lamination scheme [0/90°/0/90°/0] of the composite plate is studied. This laminated plate is supported by a set of spring components on the 8 points with stiffness values varying from to . The geometry constants of the plate layers are , . The material properties and the nondimensional foundation parameters are same as the parameters in Figure 4. From the figures, we can clearly see that the growth trend of the five parameters curves is the same as the growth trend in Figure 4. But the different place is that the effect of on the third frequency is same as the elastic spring parameters and in Figure 7. Besides, frequencies are different. It shows that the different supports have a great influence on the frequency.
As we all know, the bearing capacity of the composite lamina is directly influenced by the fibers direction and the layer’s number. Therefore, it is of great significance to study the effect of layer’s number and angles on the vibration characteristics of laminated plates on the elastic foundations. Figure 8 depicts the first three-order frequency parameters of the laminated plate on the Pasternak-type foundation. The schemes of layer’s number and angles are , where stands for the half of the layer’s number and is the angle. In this figure, there are four kinds of angle schemes which are °, 45°, 60°, and 90°. According to Figure 7, the clamped boundary constraint is simulated by setting the values of five parameters to greater than . Therefore, the elastic boundary restraint parameter , , , , and is 106. The layers of the plate have equal thickness and the material type is Material II. The geometric parameters are , . In addition, two linear foundation parameters are and . Initially, the frequencies begin to increase rapidly with the increase of layer’s number. Then, the frequencies tend to be stable until the number of layers exceeds 12. For the first and second frequencies, the values increase when the angles increase. However, this phenomenon does not apply to the third-order frequency.
The laminated plates on the elastic foundation with various multipoints supports extensively exist in engineering applications, like point supported glass curtain wall, spot-welded plate, and so on. This requires us to understand the vibration characteristics of multipoint supported plate. Figure 9 considers the multipoints supported plate resting on the Pasternak foundation. These points are evenly distributed on the four edges. The boundary restraint parameter , , , , and is 106 and the two foundation parameters are and . The geometrical dimensions and material properties are the same as Figure 8. Based on the conclusions drawn from Figure 8, the lowest four frequency parameters of the [0/45°]6 layered plate which is rested on Pasternak-type foundation against the number of points are depicted in Figure 9. In addition, we also give the of the same plate model with the equivalent uniform support as the contrast data. The number of supported points on the four edges gradually increases from 8 to 70. When the number of supported points increases, the first four frequency parameters increase firstly. Then, they tend to be stable and are similar to the frequencies of the same plate with uniform support when the number of points exceeds 48. The biggest difference of the worst case which is made is the contrast of 44 points and uniform support one being less than 0.182%. Then the biggest difference of the worst case which is made is the contrast of 48 points and uniform support one being about 0.069%. So, the natural frequencies of these point supports are finally converged to the ones of the corresponding uniform supported plate. Therefore, the multipoints support can be used to replace the uniform boundary conditions. For example, multiple bolts are tightened to simulate the uniform clamped boundary conditions in the experiment.
(a) The 1st and 2nd frequency parameters
(b) The 3rd and 4th frequency parameters
Figure 10 gives the transverse vibration displacement level on direction of isotropic plates in the frequency range of 0–400 Hz. The plate with simple support on the four endpoints is encouraged by a harmonic point force. The geometry parameters and material parameters of plate are same as Figure 5. The position of point force is at (, ), and the two observation points are at (, ) and (, ), respectively. The results obtained by FEM are given here in order to verify the accuracy of the present method. The vibration response curves at different observation points are very different. These rules are the same as those in Figure 5. It should be pointed out that the number of resonant peaks obviously increases at the observation point which is far from the position of point force.
(a) Displacement response at (, )
(b) Displacement response at (, )
Figure 11 studies the effect of elastic foundation parameters on the vibration response of the plate with 8-point elastic support. The five kinds of combinations are and , and , and , and , and and . The boundary condition of plate is . The geometry constants of the plate layers are , , and the material type is Material II. The position of point force is at (, ). The positions of the two observation points are at (, ) and (, ). Four groups of comparison curves of vibration displacement levels in the frequency range of 0–1500 Hz are given in Figure 11. It shows the vibration response with the change of Winkler-type foundation parameters in (a) and (b). With the increase of the Winkler parameter , the natural frequency increases, the displacement response wave moves back in the frequency domain, and the wave crest is attenuated. In (c) and (d) of Figure 11, the effect of Pasternak parameter on the vibration response of plate is studied. As we can see, the displacement wave moves back in the frequency domain with the increase of . Moreover, there is an interesting finding that the peak value is largest when the foundation combination is and .
(a) Displacement response at (, )
(b) Displacement response at (, )
(c) Displacement response at (, )
(d) Displacement response at (, )
At the end of this paper, the three-dimensional view models are adopted to understand the vibration characteristics of multipoints supported laminated plates on the elastic Pasternak foundations. The geometric dimensions of these moderately thick [0/45°]6 layered plates are m, m, and the material properties are the same as the material I. Five boundary restraint parameters are , , , , and . In addition, two linear foundation parameters are and , respectively. For comparison, mode shapes of the uniform supported plate on the Pasternak foundation and uniform supported plate without foundation are also given in Figure 12. As shown in Figure 12, the mode shapes tend to approach uniform boundary conditions when the number of points increases gradually. In addition, we can intuitively see the effect of the elastic foundation on the mode shapes of plate.
4. Conclusions
This paper extends an improved Fourier series solution to solve the free and forced vibration analysis of the moderately thick laminated rectangular plate on the elastic Winkler and Pasternak foundations which has various uniform and multipoints supported boundary constraints. The five displacement functions are all expressed as a series of two-dimensional Fourier series which can ignore the influence of boundary conditions. The introduction of two supplementary polynomials of the displacement functions can effectively eliminate the discontinuous or jumping phenomenon on the boundaries. The formulations are based on FSDT and the interaction of plate-foundation model and the artificial spring technique. The different boundary constraints can be easily realized by setting different elastic restraint stiffness. It can effectively simulate the boundary conditions of the plate structures in real world. The results of the present method show good agreement with the results of existing literatures and FEM. The results of free vibration for thin and moderately thick plates with various foundation parameters, different uniform boundary conditions, and multipoint supports are examined in this investigation. In addition, the lamination schemes and geometric parameters have shown a great influence on the vibration frequency of the plate. The effects of the increase of layer’s number, changes of the laying angle, and increase of the number of supported points on natural frequency of the plates are presented. The forced vibration response of plate is studied by applying an external harmonic point force. Vibration displacement response of the plate with various thicknesses, elastic boundary constraints, and elastic foundation parameters is investigated, which can provide a benchmark for the future research.
Appendix
Detailed Expressions of Matrices , , and
The specific expressions of stiffness matrix can be written as follows:
For the elastic uniform boundary conditions of the plate, the stiffness matrix can be expressed as
For the elastic multipoints supported boundary conditions of the plate, the stiffness matrix can be expressed as
The specific expressions of mass matrix M can be written as follows:
The specific expressions of mass matrix can be written as follows:
Nomenclature
| , , : | Rectangular plate dimensions |
| , , : | Linear springs stiffness |
| , , : | Plate coordinate variables |
| , : | Rotational springs stiffness |
| , , : | Middle surface displacements |
| , : | Rotations of transverse normal |
| : | Normal and shear strains |
| , : | Transverse shear strains |
| , , : | Curvature and twist changes |
| , : | Normal stresses |
| , , : | Shear stresses |
| , : | Young’s moduli with respect to principle axes of lamina |
| , , : | Shear moduli |
| , : | Poisson’s ratios |
| , , : | Normal and shear force resultants |
| : | Fiber laying angle |
| , , : | Bending and twisting moment resultants |
| , : | Transverse shear force |
| , , : | Extensional, bending, and extensional-bending stiffness coefficients |
| : | Lamina stiffness coefficients |
| : | Fourier coefficients expansions |
| : | Material coefficients |
| : | Fourier coefficients vectors |
| : | Fourier series expansions vector |
| , , , : | Stiffness, mass, coefficient, and force matrices |
| : | Shear correction factor |
| : | Nondimensional Winkler foundation parameter |
| : | Linear differential operator |
| : | Nondimensional Pasternak foundation parameter |
| : | Normal harmonic point force |
| : | Nondimensional frequency parameter |
| , : | Linear Winkler and Pasternak foundation parameters. |
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (no. 51679056) and Natural Science Foundation of Heilongjiang Province of China (E2016024).